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Gauss with Colum Pivot search

Can someone explain me (with examples) the gaussian Algorithm with Column pivot search (I'm not english, so I'm not sure if it is the right name for it).

Thanks =)
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dkloeck
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dkloeck
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1 Solution
 
GwynforWebCommented:
to explain Gauassian elimination with examples is a big job, and 30pts is not going to attract many people to this question.
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dkloeckAuthor Commented:
ok, i though it would be easy
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GwynforWebCommented:


        2y  +   z  = 1
2x  +  y  +  z   = 0
3x  + 3y +  z   = 2


0   2   1  |  1
2   1   1  |  0
3   3   1  |  2

find largest in col1 and swap row  to row1

3   3   1  |  2
2   1   1  |  0
0   2   1  |  1

divide row 1 by pivot value ie 3

1   1   1/3  |  2/3
2   1   1     |   0
0   2   1     |   1

eliminate in col 1

1    1   1/3  |  2/3
0   -1   1/3  |  -4/3
0    2   1     |   1

find largest in col2  diagonal and below and swap that row into row 2

1    1   1/3  |  2/3
0    2   1     |  1
0   -1   1/3  |  -4/3

 divide  by pivot value, ie 2

1    1   1/3  |  2/3  
0    1   1/2  |  1/2
0   -1   1/3  |  -4/3

eliminate in col 2

1    1   1/3  |  2/3
0    1   1/2  |  1/2
0    0   5/6  |  -5/6

ie

x  +  y  + 1/3z   =  2/3
        y +  1/2z   =  1/2
               5/6z   = -5/6


now back substitute

              5/6z   = 5/6   =>  z=-1

        y +  1/2z   = 1/2    =>    y = 1/2- 1/2z =1

 x  +  y  + 1/3z   =  2/3  =>x    =  2/3 - y  -1/3z  = 1-1/3-2/3 = 0

x=0  y=1  z=-1

(substituting in to orignal equations for a check shows this is correct)




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GwynforWebCommented:
 shld read            5/6z   =- 5/6   =>  z=-1
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dkloeckAuthor Commented:
it would be enough if someone gives me an example on a 3x3 System, for example:

(2  1 -2) (x)   (10)
(3  2  2)ยท(y)= (1)
(5  4  3) (z)   (4)

I think this one is positiv definit, so it can be made.
Please make it step by step with some explanation

Thanks
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dkloeckAuthor Commented:
didnt see that before ^_^
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GwynforWebCommented:
Thx for the points. I will note that there are 2 similar algorithms , one called (1) Gaussian elimation which forms a triangular matrix and backsubstitutes and (2) Gauss-Jordan that forms a unit matrix and reads the values off. (1) is faster for the computer (2) is easier by hand (ie for humans)
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